Some questions that can be asked (but not necessarily answered!) Can the fire be contained? How many time steps is required before fire is contained? How many firefighters per time step are necessary? What fraction of all vertices will be saved (burnt)? Does where the fire breaks out matter? “Smart fires”? Fire starting at more than 1 vertex? Consider different graphs. Construction of (connected) graphs to minimize damage. Complexity/Algorithmic issues
Some references 1.The firefighter problem for graphs of maximum of degree three (Finbow, King, MacGillivray, Rizzi) 2.Graph-theoretic models of spread and competition (Hartke) 3.On the firefighter problem (MacGillivray, Wang) 4.Catching the fire on grids (Fogarty) 5.Fire control on graphs (Wang, Moeller) 6.Firefighting on trees: How bad is the greedy algorithm? (Hartnell, Li) 7.On minimizing the effects of fire or a virus on a network (Finbow, Hartnell, Li, Schmeisser) 8.On designing a network to defend against random attacks of radius two (Finbow, Hartnell) 9.The optimum defense against random subversions in a network (Hartnell) 10.On minimizing the effects of betrayals in a resistance movement (Gunther, Hartnell)
Four general classes of problems 1. Containing fires in infinite grids where is the dimension.
Four general classes of problems 2.Saving vertices in finite grids of dimension 2 or 3.
Four general classes of problems 3.Firefighting on trees. Algorithmic and complexity issues.
Four general classes of problems 4.Construction of graphs that minimizes damage.
Containing fires in infinite grids L d Fire starts at only one vertex: d= 1: Trivial. d = 2: Impossible to contain the fire with 1 firefighter per time step
Containing fires in infinite grids L d d = 2: Two firefighters per time step needed to contain the fire. 8 time steps 18 burnt vertices
…… Containing fires in infinite grids L d d 3: Fact: If G is a k-regular graph, k – 1 firefighters per time step is always sufficient to contain any fire outbreak (at a single vertex) in G..….
Containing fires in infinite grids L d d 3: Fact: If G is a k-regular graph, k – 1 firefighters per time step is always sufficient to contain any fire outbreak (at a single vertex) in G. Shown: 2d – 2 firefighters per time step are not enough to contain an outbreak in L d Thus, 2d – 1 firefighters per time step is the minimum number required to contain an outbreak in L d and containment can be attained in 2 time steps.
Containing fires in infinite grids L d Theorem (Hartke): Let be a rooted graph, a positive integer, and positive integers each at least such that the following holds: 1.Every nonempty satisfies 2.For, every where satisfies 3.For, every such that satisfies
Containing fires in infinite grids L d Theorem (Hartke): Suppose that at most firefighters per time step are deployed. Then regardless of the sequence of firefighter placements. Specifically, firefighters per time step are insufficient to contain an outbreak that starts at the root vertex.
Containing fires in infinite grids L d Fire can start at more than one vertex. d = 2: Two firefighters per time step are sufficient to contain any outbreak at a finite number of vertices. d 3: For any d 3, and any positive integer, firefighters per time step is not sufficient to contain all finite outbreaks in L d. In other words, for d 3 and any positive integer, there is an outbreak such that firefighters per time step cannot contain the outbreak.
Saving vertices in finite grids G Assumptions: 1.1 firefighter is deployed per time step 2.Fire starts at one vertex Let MVS(G, v) = maximum number of vertices that can can be saved in G if fire starts at v.
Some asymptotic results Let be any vertex of Then the maximum number of vertices which can be saved by deploying one firefighter per time step with an initial outbreak at grows at most as In particular, In fact, the optimal number of vertices that can be saved given an initial outbreak at (0,0,0) in when deploying one firefighter per time step is between
Algorithmic and Complexity matters FIREFIGHTER Instance: A rooted graph and an integer Question: Is That is, is there a finite sequence of vertices of such that if the fire breaks out at then, 1.vertex is neither burning nor defended at time 2.At time no undefended vertex is adjacent to a burning vertex, and 3.At least vertices are saved at the end of time
Algorithmic and Complexity matters FIREFIGHTER is NP-complete for bipartite graphs. EXACT COVER BY 3-SETS (X3C) Instance: A set with and a collection C of 3- element subsets of Question: Does C contain an exact cover for That is, is there a sub-collection C’ C such that each element of occurs in exactly one member of C’ ?
Algorithmic and Complexity matters Suppose an instance of X3C ( C ) is given. We construct a rooted bipartite graph and a positive integer such that At least vertices of can be saved There is an exact cover of by elements of C
Algorithmic and Complexity matters for each element in C :::: CiCi CjCj For each pair of C i, C j such that ( C i C j = ) join their respective vertices ( ) by paths of length two ( ) Note that the graph is bipartite.
Algorithmic and Complexity matters :::: CiCi CjCj If has an exact cover… … save the vertices that corresponds to the subsets ( ) in the exact cover.
Algorithmic and Complexity matters :::: CiCi CjCj If at least vertices can be saved… …at most of the vertices ( ) can be saved by time … …if, at most ( ) ( + ) can be saved…
Algorithmic and Complexity matters Firefighting on Trees:
Algorithmic and Complexity matters Greedy algorithm: For each At each time step, save place firefighter at vertex that has not been saved such that weight (v) is maximized.
Algorithmic and Complexity matters GreedyOptimal = 7 = 9
number of vertices saved by optimal moves whose corresponding greedy moves performs no worse Algorithmic and Complexity matters Theorem:For any tree with one fire starting at the root and one firefighter to be deployed per time step, the greedy algorithm always saves more than ½ of the vertices that any algorithm saves. S greedy = number of vertices saved by greedy algorithm S optimal = number of vertices saved by optimal algorithm S optimal A B +
* A Algorithmic and Complexity matters GreedyOptimal S optimal B B S greedy S optimal A S optimal = S optimal + A S optimal B why was this vertex chosen on second move and not this? …because *’s ancestor has already been selected… S greedy …vertices saved by moves have already been saved by S optimal B S greedy > S optimal B S greedy > ½ S optimal
p vertices :::: … Algorithmic and Complexity matters Assume Greedy: Optimal:
Algorithmic and Complexity matters Slight modification: Suppose we are allowed to defend one vertex per time step for every burnt vertex there are at the end of the previous time step. Greedy algorithm saves at least ½ as many vertices as the optimum algorithm (Integer) Linear Programming and Dynamic Programming can be used
Algorithmic and Complexity matters 0-1 integer program for trees: subject to: for each level i for every leaf of
Algorithmic and Complexity matters for each level i for every leaf of For each vertex v r, and each descendant w of v, add the constraint Additional linear constraints can also be added to narrow the integrality gap.
Algorithmic and Complexity matters A class of trees whose can be computed in polynomial time. First, recall the definition of a perfect graph: is a perfect graph if If G is a perfect graph, we can find a maximum weight independent set of G in polynomial time.
level i Algorithmic and Complexity matters A class of trees whose can be computed in polynomial time. A rooted tree is said to be a P-tree if it does not contain the following configuration: level i+1 level i+2 No requirement for this to be an induced subgraph not a P-treea P-tree
Algorithmic and Complexity matters A class of trees whose can be computed in polynomial time. Given, let be the graph obtained from T by: 1.Adding edges joining each vertex to all its descendants 2. Adding edges joining each vertex to all vertices at the same level Now assign a weight to each vertex defined by: wt(r) = 0; wt(v) = desc(v) +1.
Algorithmic and Complexity matters A class of trees whose can be computed in polynomial time. Theorem (MacGillivray, Wang) A rooted tree is a P-tree if and only if is a perfect graph Note that an independent set in corresponds to a subset of vertices in with at most one vertex in each level. So, (T,r) is a P-tree P(T,r) is a perfect graph can find max. wt. indep set in poly. time can compute MVS(T,r) in poly. time
Some further questions to ponder… 1. For infinite graphs (like what we did for infinite grids), what is the minimum number of firefighters per time step so that only a finite number of vertices are burned? (Percolation theory?) 2. (For trees) Characterization of when the greedy algorithm is optimal. 3. Narrowing integrality gap. 4. Determination of MVP for pre-emptive vaccination. 5. Construction of networks that are resistant to attacks. 6. Can we include weight on edges to represent rate of transmission? 7. Game theory?